关于Hölder同态群及其正则性
On groups of Hölder diffeomorphisms and their regularity
摘要：
本文研究了$\mathbb R^d$的保向同构集$\mathcal D^{n,β}(\mathbb R^d)$，它不同于Hölder $C^{n,β}_0$ -映射的恒等式，其中$n \in \mathbb N_{\ge 1}$和$β\in (0,1]$ .我们证明了$\mathcal D^{n,β}(\mathbb R^d)$形成一个群，但在$\mathcal D^{n,β}(\mathbb R^d)$中的左平移一般是不连续的。然而，群$\mathcal D^{n,β-}(\mathbb R^d) := \bigcap_{α< β} \mathcal D^{n,α}(\mathbb R^d)$（具有其自然Fréchet拓扑）和$\mathcal D^{n,β+}(\mathbb R^d) := \bigcup_{α> β} \mathcal D^{n,α}(\mathbb R^d)$（具有其自然归纳局部凸拓扑）对于任何缓慢消失的连续模$ω$都是$C^{0,ω}$李群。特别地，$\mathcal D^{n,β-}(\mathbb R^d)$是一个拓扑群和所谓的半李群（具有光滑右平移）。我们证明了Hölder空间$C^{n,β}_0$是ODE闭的，在这个意义上，逐点时间依赖的$C^{n,β}_0$向量场$u$在$\mathcal D^{n,β}(\mathbb R^d)$中有唯一的流$Φ$。这特别包括所有Bochner可积函数$u \in L^1([0,1],C^{n,β}_0(\mathbb R^d,\mathbb R^d))$。对于后者和$n\ge 2$，我们表明，流图$L^1([0,1],C^{n,β}_0(\mathbb R^d,\mathbb R^d)) \to C([0,1],\mathcal D^{n,α}(\mathbb R^d))$，$u \mapsto Φ$，是连续的（甚至$C^{0,β-α}$），对于每个$α< β$。作为应用，我们证明了图像分析中相应的Trouvé群$\mathcal G_{n,β}(\mathbb R^d)$与$\mathcal D^{n,β}(\mathbb R^d)$的恒等式的连通分量是一致的.
英文摘要：
We study the set $\mathcal D^{n,β}(\mathbb R^d)$ of orientation preserving diffeomorphisms of $\mathbb R^d$ which differ from the identity by a Hölder $C^{n,β}_0$-mapping, where $n \in \mathbb N_{\ge 1}$ and $β\in (0,1]$. We show that $\mathcal D^{n,β}(\mathbb R^d)$ forms a group, but left translations in $\mathcal D^{n,β}(\mathbb R^d)$ are in general discontinuous. The groups $\mathcal D^{n,β-}(\mathbb R^d) := \bigcap_{α< β} \mathcal D^{n,α}(\mathbb R^d)$ (with its natural Fréchet topology) and $\mathcal D^{n,β+}(\mathbb R^d) := \bigcup_{α> β} \mathcal D^{n,α}(\mathbb R^d)$ (with its natural inductive locally convex topology) however are $C^{0,ω}$ Lie groups for any slowly vanishing modulus of continuity $ω$. In particular, $\mathcal D^{n,β-}(\mathbb R^d)$ is a topological group and a so-called half-Lie group (with smooth right translations). We prove that the Hölder spaces $C^{n,β}_0$ are ODE closed, in the sense that pointwise time-dependent $C^{n,β}_0$-vector fields $u$ have unique flows $Φ$ in $\mathcal D^{n,β}(\mathbb R^d)$. This includes, in particular, all Bochner integrable functions $u \in L^1([0,1],C^{n,β}_0(\mathbb R^d,\mathbb R^d))$. For the latter and $n\ge 2$, we show that the flow map $L^1([0,1],C^{n,β}_0(\mathbb R^d,\mathbb R^d)) \to C([0,1],\mathcal D^{n,α}(\mathbb R^d))$, $u \mapsto Φ$, is continuous (even $C^{0,β-α}$), for every $α< β$. As an application we prove that the corresponding Trouvé group $\mathcal G_{n,β}(\mathbb R^d)$ from image analysis coincides with the connected component of the identity of $\mathcal D^{n,β}(\mathbb R^d)$.
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